Theorem mulgnnp1 13466

Description: Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.)

Hypotheses

Ref	Expression
mulg1.b	|- B = ( Base ` G )
mulg1.m	|- .x. = (.g ` G )
mulgnnp1.p	|- .+ = ( +g ` G )

Assertion

Ref	Expression
mulgnnp1	|- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ X ) )

Proof of Theorem mulgnnp1

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> N e. NN )
2		nnuz 9684	. . . . 5 |- NN = ( ZZ>= ` 1 )
3	1, 2	eleqtrdi 2298	. . . 4 |- ( ( N e. NN /\ X e. B ) -> N e. ( ZZ>= ` 1 ) )
4		simplr 528	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> X e. B )
5		simpr 110	. . . . . 6 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> u e. ( ZZ>= ` 1 ) )
6	5, 2	eleqtrrdi 2299	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> u e. NN )
7		fvconst2g 5798	. . . . . . 7 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) = X )
8		simpl 109	. . . . . . 7 |- ( ( X e. B /\ u e. NN ) -> X e. B )
9	7, 8	eqeltrd 2282	. . . . . 6 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. B )
10	9	elexd 2785	. . . . 5 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. _V )
11	4, 6, 10	syl2anc 411	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { X } ) ` u ) e. _V )
12		simprl 529	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> u e. _V )
13		mulg1.b	. . . . . . . 8 |- B = ( Base ` G )
14	13	basmex 12891	. . . . . . 7 |- ( X e. B -> G e. _V )
15		mulgnnp1.p	. . . . . . . 8 |- .+ = ( +g ` G )
16		plusgslid 12944	. . . . . . . . 9 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
17	16	slotex 12859	. . . . . . . 8 |- ( G e. _V -> ( +g ` G ) e. _V )
18	15, 17	eqeltrid 2292	. . . . . . 7 |- ( G e. _V -> .+ e. _V )
19	14, 18	syl 14	. . . . . 6 |- ( X e. B -> .+ e. _V )
20	19	ad2antlr 489	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> .+ e. _V )
21		simprr 531	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> v e. _V )
22		ovexg 5978	. . . . 5 |- ( ( u e. _V /\ .+ e. _V /\ v e. _V ) -> ( u .+ v ) e. _V )
23	12, 20, 21, 22	syl3anc 1250	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> ( u .+ v ) e. _V )
24	3, 11, 23	seq3p1 10610	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ ( ( NN X. { X } ) ` ( N + 1 ) ) ) )
25		id 19	. . . . 5 |- ( X e. B -> X e. B )
26		peano2nn 9048	. . . . 5 |- ( N e. NN -> ( N + 1 ) e. NN )
27		fvconst2g 5798	. . . . 5 |- ( ( X e. B /\ ( N + 1 ) e. NN ) -> ( ( NN X. { X } ) ` ( N + 1 ) ) = X )
28	25, 26, 27	syl2anr 290	. . . 4 |- ( ( N e. NN /\ X e. B ) -> ( ( NN X. { X } ) ` ( N + 1 ) ) = X )
29	28	oveq2d 5960	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ ( ( NN X. { X } ) ` ( N + 1 ) ) ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ X ) )
30	24, 29	eqtrd 2238	. 2 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ X ) )
31		mulg1.m	. . . 4 |- .x. = (.g ` G )
32		eqid 2205	. . . 4 |- seq 1 ( .+ , ( NN X. { X } ) ) = seq 1 ( .+ , ( NN X. { X } ) )
33	13, 15, 31, 32	mulgnn 13462	. . 3 |- ( ( ( N + 1 ) e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) )
34	26, 33	sylan 283	. 2 |- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) )
35	13, 15, 31, 32	mulgnn 13462	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) )
36	35	oveq1d 5959	. 2 |- ( ( N e. NN /\ X e. B ) -> ( ( N .x. X ) .+ X ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ X ) )
37	30, 34, 36	3eqtr4d 2248	1 |- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   {csn 3633   X. cxp 4673   ` cfv 5271  (class class class)co 5944   1c1 7926   + caddc 7928   NNcn 9036   ZZ>=cuz 9648   seqcseq 10592   Basecbs 12832   +g cplusg 12909  ._gcmg 13455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-2 9095  df-n0 9296  df-z 9373  df-uz 9649  df-seqfrec 10593  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-minusg 13336  df-mulg 13456

This theorem is referenced by:  mulg2  13467  mulgnn0p1  13469  mulgnnass  13493  gsumfzconst  13677